#21 Control Theory #21 - Time-Domain Specifications

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Second-order response

   Corrected solution notes. The audio is the original recording and may contain errors or incomplete phrases. Where they differ, use the corrected written solution.

   Use the standard stable, zero-free second-order model with natural frequency ωn and damping ratio ζ.

   Its normalized unit-step shape depends on these parameters; zeros, delays and additional modes require separate analysis.

   Narration transcript

   When we apply a step input to a second order system, the output follows a characteristic curve. It rises, may overshoot the target, oscillates, and eventually settles. The shape of this curve tells us everything about the system's performance. Today we learn how to read and quantify that shape.

2. 2. Rise time

   Here rise time means the interval between the first 10% and 90% crossings of the final value.

   Keep this convention explicit when comparing another lesson using a 100% crossing.

   Narration transcript

   The first parameter is rise time, t r. It measures how quickly the system responds. We define it as the time for the output to go from ten percent to ninety percent of its final value. A smaller rise time means a faster system. Watch the curve: the steeper the initial climb, the shorter the rise time.

3. 3. Peak and overshoot

   For 0<ζ<1, damped frequency is ωd=ωn√(1−ζ²), and peak time is π/ωd.

   The overshoot symbol is Mp. Its fractional value is exp(−πζ/√(1−ζ²)); multiply by 100 for a percentage.

   Narration transcript

   Next, peak time t p is the time to reach the first peak. The overshoot, M p, is how far the response exceeds the final value, expressed as a percentage. An overshoot of twenty percent means the peak reaches one point two times the final value. High overshoot means the system is too oscillatory. The formula for percent overshoot is e to the negative pi zeta over the square root of one minus zeta squared, times one hundred.

4. 4. Settling time

   The settling time is the last entry into a specified band around the final value.

   The common 2% estimate is 4/(ζωn); it is an approximation, not an exact last-crossing calculation.

   Narration transcript

   Settling time t s is the time for the response to stay within a band around the final value. We typically use a two percent or five percent band. Once the oscillations decay enough to stay inside this band, the system has settled. The formula is approximately four divided by zeta times omega n, for the two percent criterion.

5. 5. Read the four specifications

   Rise time, peak time, overshoot and settling time summarize useful transient features.

   Four measured numbers do not uniquely determine an arbitrary system or prove a second-order model.

   Narration transcript

   Let us see all four parameters together on the same curve. Rise time captures the initial speed. Peak time marks the first maximum. Overshoot measures the excess above the target. And settling time tells us when the oscillations become negligible. These four numbers fully characterize the transient behavior of a second order system.

6. 6. Changing damping

   Holding ωn fixed changes both speed and oscillation as ζ varies.

   Standard underdamped responses have overshoot; critically damped and overdamped zero-free responses do not. Choose damping from the actual design constraints.

   Narration transcript

   Now the key insight: all these parameters depend on just two numbers. Zeta, the damping ratio, and omega n, the natural frequency. Watch what happens as we change zeta. With low zeta, the system is fast but oscillatory, with large overshoot. With high zeta, the system is slow but smooth, with no overshoot. The sweet spot for most designs is zeta between zero point four and zero point eight.

7. 7. Pole-location interpretation

   Standard underdamped poles are −ζωn±jωn√(1−ζ²).

   Approximate settling constraints bound the real part; overshoot constraints bound damping ratio. Rise-time bounds require a stated model and approximation.

   Narration transcript

   We can translate time domain specifications directly onto the s plane. A settling time requirement defines a vertical boundary. The poles must be far enough to the left. An overshoot requirement defines a wedge shaped region. The poles must be inside a sector with angle determined by zeta. A rise time requirement pushes poles further from the origin. The intersection of all these regions gives the acceptable pole locations.

8. 8. Numerical example

   With ζ=0.4 and ωn=5 rad/s, ωd≈4.5826 rad/s, peak time≈0.6856 s, and overshoot≈25.383%.

   The 2% settling estimate is 2 s. The source rise-time estimate 1.4/ωn gives 0.28 s; exact 10–90% rise time must be computed from its two crossings.

   Narration transcript

   Let us work a concrete example. Given a system with zeta equals zero point four and omega n equals five radians per second. Rise time: approximately one point four divided by omega n, which gives zero point two eight seconds. Peak time: pi divided by omega n times the square root of one minus zeta squared, which gives zero point six eight seconds. Overshoot: using the formula, we get approximately twenty five point four percent. Settling time: four divided by zeta times omega n, which gives two seconds. These numbers tell us: fast response, moderate overshoot, settles in two seconds.

9. 9. Review

   Preserve the Mp notation and distinguish exact model formulas from engineering estimates.

   Verify model assumptions and measurement conventions before translating time-domain specifications into pole constraints.

   Narration transcript

   To summarize. Four parameters describe the transient response: rise time, peak time, percent overshoot, and settling time. They all come from zeta and omega n. Low zeta means fast but oscillatory. High zeta means slow but smooth. The s plane connects specifications to pole placement. In the next lesson, we study steady state error and how system type determines the error for different input signals.

Source video: Control Theory #21 - Time-Domain Specifications (4:47)